A nonlinear model for a towed flexible cylinder

Abstract

In this paper, a weakly nonlinear equation of motion is derived for the dynamics of a towed, neutrally buoyant flexible slender cylinder. The cylinder is terminated by end-pieces at its two ends and is fastened via a massless towrope to a support rigidly fixed upstream. The motions are considered to take place in a horizontal plane. The equation of motion is obtained via Hamilton's principle after obtaining the Lagrangian of the system and the virtual work associated with the fluid dynamic forces. For convenience, the fluid-related forces are derived separately: inviscid hydrodynamic forces are modelled by an extension of Lighthill's slender-body work to third-order accuracy, and the viscous forces are derived to the same accuracy, by elaboration of Taylor's expressions. The Galerkin method is used to discretize the equation of motion with the free–free Euler–Bernoulli beam eigenfunctions, and the resulting set of first-order equations are solved numerically using a time-integration solver. Numerical results are obtained to illustrate the typical dynamical behaviour of a towed flexible cylinder, generally confirming experimental observations and linear theory predictions made in the past. Also, the effect of the towrope-length to cylinder-length ratio and the effect of the tail end-piece shape on the dynamics and stability of the system are investigated. The results confirm that for a towed flexible cylinder, if the tail end-piece is not blunt and the towrope not too short, both rigid-body and flexural instabilities may develop as the flow velocity is increased. The former occur at low flow velocities, in the form of oscillatory and then static instabilities, whereas the latter generally occur at higher flow velocities in the form of second- and then third-mode flutter. Moreover, it is found that the system becomes less stable and is subject to larger deformations if the towrope is longer. On the other hand, making the tail end-piece sufficiently blunt can effectively suppress all the instabilities; the system then remains stable for apparently all towing speeds.